Nonlinear Trajectory Optimization with Path Constraints Applied to Spacecraft Reconfiguration Maneuvers by Ian Miguel Garcia S.B. Aeronautics and Astronautics, Massachusetts Institute of Technology, 2003 S.B. Mathematics, Massachusetts Institute of Technology, 2003 Submitted to the Department of Aeronautics and Astronautics in partial fulfillment of the requirements for the degree oSCUSTSiS OF TECHNOLOGY Master of Science in Aeronautics and Astronautics JUN 2 3 2005 at the LIBRARI MASSACHUSETTS INSTITUTE OF TECHNOLOG June 2005 @ Massachusetts Institute of Technology 2005. All rights reserved. A uthor .......... ... .. ............... ............. Department of Aeronautics and Astronautics May 20, 2005 C ertified by ................... ......... Jonathan P. How 4ssociate Professor Jlhesis Supervisor Accepted by.............. Jaime Peraire Professor of Aeronautics and Astronautics Chair, Committee on Graduate Students AERO Nonlinear Trajectory Optimization with Path Constraints Applied to Spacecraft Reconfiguration Maneuvers by Ian Miguel Garcia Submitted to the Department of Aeronautics and Astronautics on May 20, 2005, in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics Abstract Future space assembly and science missions (e.g., Terrestrial Planet Finder) will typ- ically require complex reconfiguration maneuvers involving well coordinated 6 DOF motions of the entire fleet of spacecraft. The motions must also satisfy constraints such as collision avoidance and pointing restrictions on some of the instruments. This problem is particularly difficult due to the nonlinearity of the attitude dynamics and the non-convexity of some of the constraints. The coupling of the positions and atti- tudes of the N spacecraft by some of the constraints adds a significant complication because it requires that the trajectory optimization be solved as a single 6N DOF problem. This thesis presents a method to solve this problem by first concentrating on finding a feasible solution, then developing improvements to it. The first step is posed as a path planning problem without differential constraints and solved using Rapidly-exploring Random Trees (RRT's). The improvement step is posed as a fea- sible nonlinear optimization problem and solved by an iterative optimization similar to a sequential linear programming method. The primary contribution of the thesis is an improvement to the basic RRT algorithm that replaces the connection step with a more complex function that takes into account local information about the con- straints. Two functions are proposed, one based on artificial potential functions, and another based on a random search. The results show that the new RRT with either of these connection functions is faster and more reliable for a difficult sample problem. The combination of an RRT with the new connection functions, and the improvement step, is also demonstrated on several challenging spacecraft reconfiguration problems with up to 16 spacecraft (96 DOF). The solution technique is shown to be robust and with computation times fast enough to embed in a real-time optimization algorithm as part of an autonomous onboard control system. Thesis Supervisor: Jonathan P. How Title: Associate Professor 3 4 Acknowledgments I would like to thank my advisor, Prof. Jonathan How, for sharing with me the questions and insight that made this work possible, and for going out of his way to help me complete it. I want to express my gratitude as well to the members of the research group who, I have to honestly say, are a bunch of funny, smart and helpful people with whom working has been real pleasure. I would also like to thank the MIT Department of Aeronautics and Astronautics for awarding me with a Departmental Fellowship which helped support part of this work. To Papd, Mami and Javi This work was funded under NASA Grant NAG5-10440 and Cooperative Agreement NCC5-729 through the NASA GSFC Formation Flying NASA Research Announce- ment. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Aeronautics and Space Administration. 5 6 Contents Abstract 3 Acknowledgements 5 Table of Contents 6 List of Figures 10 List of Tables 14 1 Introduction 17 1.1 Nonlinear Trajectory Optimization . ........ ......... 17 1.2 Spacecraft Formation Reconfiguration with Position and Attitude Con- straints . ........ ........ ........ ....... ... 20 1.3 Solution Approach to Hard Trajectory Optimization Problems ... 22 1.4 An Improved Planner ...... ............. 23 1.5 Trajectory Optimization with Discontinuous Constraints or Cost . 24 1.6 O verview ...... ....... ...... ...... ...... 25 2 Trajectory Optimization for Satellite Reconfiguration Maneuvers 27 2.1 The Spacecraft Reconfiguration Problem Formulation ...... 28 2.2 The Solution Approach ..... ...... ...... ...... ... 30 2.2.1 The Planner ....... ........ ....... 30 2.2.2 The Sm oother ... ..... ..... .... ..... ..... 32 2.3 Exam ples .... ...... ...... ..... ...... ...... 37 7 2.3.1 Example: Simple Maneuver ................... 39 2.3.2 Example: Coupled Maneuver .................. 39 2.3.3 Example: Four Vehicles ................. .... 44 2.4 Comparison with a Nonlinear Optimal Control Technique .... ... 44 2.4.1 Problem Formulation with DIDO .. ........... ... 44 2.4.2 Reconfiguration Problem for Different Initial Solutions and Prob- lem Sizes ....... ....... ....... ....... .. 45 2.5 Attitude Maneuver with 1 Spacecraft ..... ........ ..... 54 2.6 Sum m ary .. ........ ......... ........ ...... 55 3 The New Connection Functions 57 3.1 Introduction ................................ 57 3.2 The New Connection Functions ............ ......... 59 3.2.1 The Potential Connection Function ............... 60 3.2.2 The Random Connection Function .. ............. 62 3.3 Illustration of Connection Functions with Simple 2D Path Planning Exam ple .................................. 63 3.3.1 Comparison of Coverage ..................... 76 3.4 Simulation Results with the Spacecraft Reconfiguration Problem .. 78 3.4.1 Three Spacecraft ......................... 80 3.4.2 Change of Formation Shape ................... 80 3.4.3 Formation Rotation with 4 Spacecraft ............. 81 3.4.4 Formation Reflection with 4 Spacecraft ............. 81 3.4.5 Comparison of Computation Time ............... 81 3.4.6 New Results with Larger Configurations ............ 81 3.4.7 Crossing at the center .. ........ ........ .... 82 3.4.8 Rotation of Star Configuration ..... ....... ...... 82 3.4.9 Random Diagonal Configuration to Star Configuration ... 82 3.5 Sum m ary ....... ............. ............ 83 8 4 DIDO and NTG Applied to Trajectories with Discontinuous Con- straints 87 4.1 Introduction ....... ................................ 87 4.2 The Nonlinear Problem ................ 88 4.3 Problems with Known Path Discontinuities ............... 91 4.4 Multi-phase Problem in DIDO and NTG . ............... 94 4.5 The Sensor Problem ........................... 95 4.6 Simulation of the Sensor Problem .................... 98 4.7 The Weighted Shortest Path ....................... 99 4.8 Simulation of the Weighted Shortest Path Problem .......... 101 4.9 Sum m ary ............ ............. ........ 104 5 Conclusions and Future Work 107 5.1 C onclusions ....... ............ ........... .. 107 5.2 Future Work . ........................ ....... 109 A UAV 2D path planning with DIDO and NTG 111 A.1 The UAV Problem with Obstacles ............. ....... 111 A.2 Implementation with DIDO ....................... 112 A.3 Implementation with NTG ............ ............ 114 Bibliography 115 9 10 List of Figures 1.1 NASA Terrestrial Planet Finder mission . 19 1.2 NASA Stellar Imager mission ......... ............ 19 1.3 DARPA Orbital Express mission ............... ..... 20 1.4 The formation reconfiguration problem ....... .......... 21 1.5 The best trajectory minimizes the length over the high risk area .. 25 2.1 Example: Simple maneuver. Simple translation and rotation with sun avoidance constraint. ..... ............ .......... 38 2.2 Locus of "instrument" vector is tight against sun avoidance constraint 38 2.3 Cosine of angle between "instrument" vector and sun vector ..... 38 2.4 Example: Coupled maneuver ............ ........... 40 2.5 Same as Figure 2.4, solution before the smoothing ... ........ 40 2.6 Locus of instrument in spacecraft 2 with sun avoidance constraint . 41 2.7 Locus of vector pointing from spacecraft 1 to spacecraft 2 . ...... 41 2.8 Locus of vector pointing from spacecraft 2 to spacecraft 1 .... ... 41 2.9 Cosines of angles between ranging device vector and relative position of both spacecrafts ....... ............................ 42 2.10 Rotate tetrahedral configuration 90" degrees around Y axis. Pointing constraints remain as before. ................... .... 43 2.11 Plan and smoothed trajectories for Example A ............. 46 2.12 Plan and smoothed trajectories for Example B ... .......... 47 2.13 Performance of DIDO for Example A ............. ..... 49 2.14 Performance of DIDO for Example B ............. ..... 49 11 2.15 Performance of DIDO for Example C (with stay-inside constraint) 51 2.16 Performance of DIDO for Example D (with stay-outside constraint) 51 2.17 Performance of DIDO for Example E (with both stay-inside and stay- outside constraints) ... ..... ...... ..... ...... 52 2.18 Success ratio for DIDO for the examples shown here . ...... 53 2.19 Locus of Telescope